tasks in probability theory

nom_1.py

# -*- coding: utf-8 -*-


from random import random
from math import log


def main() :
    lmbd = float(input())
    eto = random()
    print(-log(eto) / lmbd)
    
    
if __name__ == "__main__" :
    main()

nom_2.py

# -*- coding: utf-8 -*-


from random import random
from math import *


def transformation_box_muller(a1, a2):
    return sqrt(2*log(1/a1))*sin(2*pi*a2)


def normal_distribution(x, sigma, mu):
    return (1/(sigma*sqrt(2*pi)))*exp(-(x-mu)**2/(2*sigma**2))


def main():
    a1 = random()
    a2 = random()
    x = transformation_box_muller(a1, a2)
    print(normal_distribution(x, 1, 0))


if __name__ == "__main__":
    main()

nom_3.py

# -*- coding: utf-8 -*-


from math import *
import matplotlib.pyplot as plt


def BuildGraph(name, dataX, dataY) :
    graph = plt.figure()
    f = graph.add_subplot(111)
    f.plot(dataX, dataY)
    f.grid(True)
    f.set_xlabel("n")
    f.set_ylabel("p_n")
    f.set_title(name)
    plt.savefig(name + ".png")
    plt.show()
    
    
def Cc(m, N) :
    return factorial(N)/(factorial(N-m)*factorial(m))
    
    
def FirstFunc(p, N) :
    arr_pn = []
    arr_n = []
    
    for n in range(0, N+1) :
        pn = Cc(n, N) * (p**n) * ((1-p)**(N-n))
        arr_pn.append(pn)
        arr_n.append(n)
        
    BuildGraph("FirstFunc", arr_n, arr_pn)
    
    
def SecondFunc(p, N) :
    arr_pn = []
    arr_n = []
    
    for n in range(0, N+1) :
        pn = (((N*p)**n) / factorial(n)) * exp(-N*p)
        arr_pn.append(pn)
        arr_n.append(n)
        
    BuildGraph("SecondFunc", arr_n, arr_pn)
    

def ThirdFunc(p, N) :
    arr_pn = []
    arr_n = []
    
    sigma_sqr = p * (1 - p)
    dxn = 1/sqrt(N)
    for n in range(0, N+1) :
        xn = (n - N*p)/sqrt(N)
        pn = (1 / sqrt(2*pi*sigma_sqr)) * exp(-xn**2/(2*sigma_sqr)) * dxn
        arr_pn.append(pn)
        arr_n.append(n)
        
    BuildGraph("ThirdFunc", arr_n, arr_pn)


def main() :
    p = float(input("p: ")) ## p = 0.5
    N = int(input("N: ")) ## N = 50
    
    FirstFunc(p, N)
    SecondFunc(p, N)
    ThirdFunc(p, N)
    

if __name__ == "__main__" :
    main()

nom_4&5.py

# -*- coding: utf-8 -*-


from math import *
import matplotlib.pyplot as plt
from random import random
from scipy.special import gamma, hyp2f1
import numpy as np


def BuildGrpah(hist_data, graph_data) :
    bins = np.linspace(0, 2.2, 50)
    plt.hist(hist_data, bins, density=True)
    plt.plot(graph_data[0], graph_data[1])
    plt.savefig("keks.png")
    plt.show()
    

def Nomber4(a, b) :
    
    def Distribution(a, b) :
        return random()**a + random()**b
    
    
    data = []
    for i in range(100000) :
        g = Distribution(a, b)
        data.append(g)
        
    return data
    
    
def Nomber5(a, b) :
    
    def P1(x) :
        return (x**(1/a + 1/b - 1)) * (gamma(1+1/a)*gamma(1+1/b) / gamma(1/a+1/b))
    
    
    def P2(x) :
        A = hyp2f1(1/a, 1-1/b, 1/a+1, 1/x) - (x-1)**(1/a)*hyp2f1(1/a, 1-1/b, 1/a+1, 1-1/x)
        return A * x**(1/b-1)/b
 
 
    def Distribution(x) :
        if -0 <= x <= 1 :
            return P1(x)
        return P2(x)
    
    
    dataX, dataY = [], []
    x = 0
    while x < 2 :
        p = Distribution(x)
        dataX.append(x)
        dataY.append(p)
        x += 0.001

    return [dataX, dataY]
    
    
def main() :
    a = float(input("a: ")) #a = 0.5
    b = float(input("b: ")) #b = 0.1
    hist_data = Nomber4(a, b)
    graph_data = Nomber5(a, b)
    BuildGrpah(hist_data, graph_data)
    
    
if __name__ == "__main__" :
    main()